Bristol 6.-8.
January 2005

This is the first of a series of meetings (supported by the British Academy)
of the research group in Logical Methods in Epistemology,
Semantics and Philosophy of Mathematics which aims to investigate,
and implement, the use of mathematical and technical logical apparatus
in philosophical fields, in particular, but not exclusively, philosophical
logic, theories of truth, and the philosophy of mathematics. The first
of these will take place at the University of Bristol Mathematics Department.

Participation is free, but we ask you to inform us at least by 20th December
if you plan to attend the meeting:

volker.halbach at philosophy.oxford.ac.uk
P.Welch at bristol.ac.uk

Please replace "at" by the usual symbol.

Venue

All talks will be in the School of Mathematics in Seminar Room SM3, which
is on the 1st Floor relative to the Main entrance on University Walk.

There is a direct coach service from Heathrow to Bristol. On the National
Express website you make make reservations for tickets. It is much
cheaper to buy a return ticket than to buy to single tickets. Bristol
also has an airport.

From the Train Station you can take the No. 8 or 8A Bus to the Clifton
Down Road/Princess Victoria Street stop. From there the Rodney Hotel is
about 80ms.

Note for Speakers

There will be an Overhead Slides Projector, Computer Projector, and White
Boards available.

Abstracts

Branden Fitelson: A User-Friendly Decision
Procedure for the Probability Calculus, with Applications to Bayesian
Confirmation Theory

A general mechanical procedure for reasoning about the probability calculus
is presented. The procedure involves (1) a translation from probability
calculus into the theory of real closed fields (TRCF), and (2) an application
of a recent implementation of the CAD procedure for TRCF. The procedure
is then used to solve various problems in Bayesian confirmation theory
(some of which were open). Some issues of computational complexity and
problem size will also be discussed. All necessary technical, historical,
and philosophical background will be provided during the talk.

Michael Glanzberg: Contexts and Unrestricted
Quantification

Much of the debate over the possibility of ‘absolutely unrestricted
quantification’—quantification over a domain of ‘absolutely
everything’—has centered on the paradoxes. To some, such as
myself, careful consideration of the paradoxes shows absolutely unrestricted
quantification to be impossible. But those of us who hold this sort of
view face a challenge. For even if we establish on such general grounds
that absolutely unrestricted quantification is impossible, we have still
to account for the ways that our prima facie unrestricted quantifiers
really function. This challenge is made all the more pressing, as Timothy
Williamson has recently argued, by the appearance that some applications
of quantifiers require them to be absolutely unrestricted.

This paper takes up these challenges, by presenting a contextualist approach
to unrestricted quantification. It argues that even our widest, syntactically
unrestricted quantifiers, are subject to a special kind of contextual
domain restriction. However, it also argues that this is not an instance
of the ordinary sorts of contextual domain restrictions which apply to
our uses of restricted quantifiers. Instead, our widest quantifiers are
subject to a distinct sort of context dependence, which I label ‘extraordinary
context dependence’. This paper argues that though unusual, extraordinary
context dependence is possible, and sketches how it arises. It goes on
to show that the rules governing extraordinary context dependence ensure
that the domain of any maximal quantifier in a context functions nearly
enough as if it were absolutely unrestricted. Because of this, the paper
argues, our maximal quantifiers are suitable for their intended applications,
in spite of being subject to extraordinary contextual restrictions.

Jeff Ketland: Speed-Up, Indispensability and
Unfeasibility

A standard argument against nominalism is the indispensability argument,
which points out that our scientific understanding of the world is thoroughly
mathematicized. Most examples thus far examined in the literature involve
very abstract mathematics, usually from theoretical physics. I wish to
discuss the application of mathematics to very simple problems of logical
validity. The topic of applying mathematics to problems of logical validity
has not been widely discussed. There is a paper by Hartry Field, "Is
Mathematical Knowledge Just Logical Knowledge?" (1985) and some papers
by George Boolos, concerning speed-up. Nominalists seem to think that
the nominalizability of finite cardinality statements like "There
are n Fs" represents a success for their programme. I argue that
even this is a failure. Consider the inference PHP(100) (thanks to my
wife for the example):

There are 101 dalmatians
There are 100 food bowls
Each dalmatian feeds from exactly one food bowl
So, at least two dalmatians share the same food bowl.

We all know that this is valid, but how? I call such logically valid
inferences "quasi-arithmetic". The validity of PHP(100) is an
example of a quasi-arithmetic "validity fact". Simple combinatorial
mathematics is indispensable for seeing that PHP(100) is valid. The reason
is that a direct logical verification for this inference would fill a
3,000 page book. In fact, we know that PHP(n) is valid for all n, and
thus that PHP(100) is valid. But the nominalist rejects these assumptions
about numbers, finite set and functions as false. So why does a false
theory of numbers, sets, etc., yield true classification of such validity-facts?
This is nothing short of a miracle! And if we reject miracles, then we
should reject nominalism too. Postulating numbers, sets and functions
is indispensable for seeing that certain valid inference are indeed valid.

This "speed-up argument against nominalism" is briefly sketched
in a short paper "Some More Curious Inferences" (Analysis, Jan
2005). A forthcoming paper by John Burgess ("Protocol Sentences for
Lite Logicism" on Burgesss webpage) contains a similar argument.

James Ladyman: Mathematical Structuralism and
the Identity of Mathematical Objects

Mathematical structuralism comes in various varieties and faces many
problems. In this paper I offer a solution to two problems that have been
raised for non-eliminative structuralism about mathematical objects of
the kind that has been defended by Stewart Shapiro. The first concerns
the truth-values of identity statements concerning mathematical objects
in different structures, for example, 'the natural number 1 is identical
to the real number 1'. The second concerns the violation of the principle
of the identity of indiscernibles to which the non-eliminative structuralist
seems committed.

Carnap's Aufbau is usually regarded as a famous - perhaps even notorious
- failure. I want to show that certain parts of the Aufbau programme can
actually be saved and be put to work. However, in order to do so, Carnap's
original intentions have to be lowered and various problems that affect
the original Aufbau system have to be solved or circumvented. In my talk
I will give a sketch of a new constitution system and I will try to outline
how it addresses the well-known difficulties concerning (i) the constitutional
basis, (ii) quasi-analysis, (iii) dimensionality, (iv) holism and theoretical
terms, (v) disposition terms, and (vi) structuralism in an Aufbau-like
setting.

Albert Visser: Tarski's Result on the Undefinability
of Truth and the Category of Interpretations

In this talk, we discuss the object-/metalanguage distinction and Tarski's
theorem on the undefinability of truth in the context of the category
of interpretations. This category offers the proper framework for the
study the interaction between truth and translation. We explicate the
notion of object-/metalanguage pair as an arrow in the category of interpretations.
In these terms,Tarski's result means that certain arrows cannot be `restricted'.

We apply the framework to discuss a construction of Orey Sentences of
a theory T, i.e. sentences such that both T+A and T+¬A are interpretable
in T. Moreover, we prove that arithmetical theories extending PA cannot
be bi-interpretable with set theories extending ZF. This shows that the
difference between arithmetical theories and set theories extends below
the surface constituted by an arbitrary choice of language. At the same
time, the result illustrates that mutual interpretability differs from
bi-interpretability, since all set theories extending ZF are mutually
interpretable with an arithmetical theory extending PA.